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Physical Review Letters

Conformal Invariance of the Ising Model in Three Dimensions

Youjin Deng, Henk W. J. Blöte

Published Apr 30, 2002• 23 citations
Publisher

Abstract

We investigate a critical Ising-like model in the curved geometry ${S}^{2}\ifmmode\times\else\texttimes\fi{}{R}^{1}$ obtained by a conformal mapping of the infinite 3D space ${R}^{3}$. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as ${X}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5178(12)$ and ${X}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.423(19)$, respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.

DOI
10.1103/physrevlett.88.190602

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Universal amplitude-exponent relation for the Ising model on sphere-like lattices

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•Jan 1, 2000•13 citations
We investigate a critical Ising-like model in the curved geometry S2\ifmmode×\else\texttimes\fiR1{S}^{2}\ifmmode\times\else\texttimes\fi{}{R}^{1}S2\ifmmode×\else\texttimes\fiR1 obtained by a conformal mapping of the infinite 3D space R3{R}^{3}R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as Xh=0.5178(12){X}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5178(12)Xh​=0.5178(12) and Xt=1.423(19){X}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.423(19)Xt​=1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.
Using a continuous cluster Monte Carlo algorithm, we investigate the critical three-dimensional Ising model in its anisotropic limit. From the ratio of the magnetic correlations in the strong- and the weak-coupling directions, we determine the length ratio relating the isotropic Ising model and the anisotropic limit. On this basis, we simulate the critical Ising model on a spherocylinder S2 x R1, i.e., a curved geometry obtained from a conformal mapping of the infinite space R3. From correlation lengths along the spherocylinder, combined with the prediction of conformal invariance, we estimate the magnetic and thermal scaling dimensions as X(h)=0.5182(6) and X(t)=1.419(7), respectively. The behavior of the Binder cumulant is also determined in the limit of an infinitely long spherocylinder. Next, free boundary conditions are imposed on the equators of the spherocylinder, and thus the geometry S1 x S+ x R1 is obtained. The surface magnetic scaling dimension is estimated as X(s)(h)=1.263(5). The consistency of the aforementioned estimations and existing results confirms that the three-dimensional Ising model is conformally invariant. Further, the precision of these results reveals that, as in two dimensions, conformal mappings provide a powerful tool to investigate critical phenomena. With the continuous cluster algorithm, we also perform simulations of systems inside a conventional solid cylinder. The surface magnetic correlation length differs, within the estimated error margin, by a factor pi/2 from that along a half spherocylinder S1 x S+ x R1 with the same radius.
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the critical behavior of the 3d model. The fractal dimension and the winding angle statistics of the perimeter and external perimeter of the geometric spin clusters at the critical point suggest that, if conformally invariant in the scaling limit, they can be described by the theory of Schramm-Löwner evolution (SLE κ ) with diffusivity of κ=5 and 16/5, respectively, putting them in the same universality class as the interfaces in 2d tricritical Ising model. It is also found that the Fortuin-Kasteleyn (FK) clusters associated with the cross-sections undergo a nontrivial percolation transition, in the same universality class as the ordinary 2d critical percolation.
We formulate conformal mappings between an infinite plane and a spheroid, and one between a semi-infinite plane and a half spheroid. Special cases of the spheroid include the surface of an infinitely long cylinder, of a sphere, and of a flat disc. These mappings are applied to the critical Ising model. For the case of the sphere and the flat disc, we derive analytical expressions for the second and the fourth moments of the magnetization density, and thus for the Binder cumulant. Next, we investigate Ising models on spheroids and half spheroids by means of a continuous cluster Monte Carlo method for simulations in curved geometries. Fixed and free boundary conditions are imposed for half spheroids. The Monte Carlo data are analyzed by finite-size scaling. Critical values of the Binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above. At criticality, we also sample two- and one-point correlation functions on spheroids on half spheroids, respectively. The magnetic and temperature scaling dimensions, as determined from the Monte Carlo data and the theory of conformal invariance, are in good agreement with exact results.
Conformal field theory predicts finite-size scaling amplitudes of correlation lengths universally related to critical exponents on sphere-like, semi-finite systems S d − 1 × of arbitrary dimensionality d . Numerical studies have up to now been unable to validate this result due to the intricacies of lattice discretisation of such curved spaces. We present a cluster-update Monte Carlo study of the Ising model on a three-dimensional geometry using slightly irregular lattices that confirms the validity of a linear amplitude-exponent relation to high precision.
The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.18(2)$, $\ensuremath{\beta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.238(5)$, and $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.36(3)$. Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.
(received; accepted) PACS. 75.10Hk – Classical spin models. PACS. 75.40Mg – Numerical simulation studies. PACS. 11.25Hf – Conformal field theory, algebraic structures. Abstract. – Conformal field theory predicts finite-size scaling amplitudes of correlation lengths universally related to critical exponents on sphere-like, semi-finite systems S d−1 ×R of arbitrary dimensionality d. Numerical studies have up to now been unable to validate this result due to the intricacies of lattice discretisation of such curved spaces. We present a cluster-update Monte Carlo study of the Ising model on a three-dimensional geometry using slightly irregular lattices that confirms the validity of a linear amplitude-exponent relation to high precision. Introduction. – The observation that the symmetry of systems at a critical point goes beyond scale invariance emerging from the divergence of correlation lengths, and additionally comprises rotational and translational as well as inversional invariance has greatly enlarged the scope of exact results that can be derived for such systems on field-theory grounds. While customary scaling theory and the real-space renormalisation ansatz [1, 2] give rise to general